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61.
A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G1, G2 ∈ P there exists a graph G ∈ P containing both G1 and G2 as subgraphs. Let H be any given graph on vertices v1, . . . , vn, n ≥ 2. A graph property P is H-factorizable over the class of graph properties P if there exist P 1 , . . . , P n ∈ P such that P consists of all graphs whose vertex sets can be partitioned into n parts, possibly empty, satisfying: 1. for each i, the graph induced by the i-th non-empty partition part is in P i , and 2. for each i and j with i = j, there is no edge between the i-th and j-th parts if vi and vj are non-adjacent vertices in H. If a graph property P is H-factorizable over P and we know the graph properties P 1 , . . . , P n , then we write P = H [ P 1 , . . . , P n ]. In such a case, the presentation H[ P 1 , . . . , P n ] is called a factorization of P over P. This concept generalizes graph homomorphisms and (P 1 , . . . , P n )-colorings. In this paper, we investigate all H-factorizations of a graph property P over the class of all hered- itary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization. 相似文献
62.
Daniel Král’ Jan Kratochvíl Andrzej Proskurowski Heinz-Jürgen Voss 《Discrete Applied Mathematics》2006,154(4):660-672
A mixed hypergraph is a triple (V,C,D) where V is its vertex set and C and D are families of subsets of V, called C-edges and D-edges, respectively. For a proper coloring, we require that each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The feasible set of a mixed hypergraph is the set of all k's for which there exists a proper coloring using exactly k colors. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of the tree.We prove that feasible sets of mixed hypertrees are gap-free, i.e., intervals of integers, and we show that this is not true for precolored mixed hypertrees. The problem to decide whether a mixed hypertree can be colored by k colors is NP-complete in general; we investigate complexity of various restrictions of this problem and we characterize their complexity in most of the cases. 相似文献
63.
The most famous open problem involving domination in graphs is Vizings conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal 相似文献
64.
Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant
double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is
circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency
4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices.
The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported
by Com2MaC-KOSEF (R11-1999-054) in Korea. 相似文献
65.
66.
Symmetry of graphs has been extensively studied over the past fifty years by using automorphisms of graphs and group theory which have played and still play an important role for graph theory, and promising and interesting results have been obtained, see for examples, [L.W. Beineke, R.J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, London, 2004; N. Biggs, Algebraic Graph Theory, Cambridge University Press, London, 1993; C. Godsil, C. Royle, Algebraic graph theory, Springer-Verlag, London, 2001; G. Hahn, G. Sabidussi, Graph Symmetry: Algebraic Methods and Application, in: NATO ASI Series C, vol. 497, Kluwer Academic Publishers, Dordrecht, 1997]. We introduced generalized symmetry of graphs and investigated it by using endomorphisms of graphs and semigroup theory. In this paper, we will survey some results we have achieved in recent years. The paper consists of the following sections.
- 1. Introduction
- 2. End-regular graphs
- 3. End-transitive graphs
- 4. Unretractive graphs
- 5. Graphs and their endomorphism monoids.
Keywords: Graph; Endomorphism; Monoid; Generalized symmetry; End-regular; End-transitive; Unretractive 相似文献
67.
《Optimization》2012,61(3):597-624
Some scheduling problems induce a mixed graph coloring, i.e., an assignment of positive integers (colors) to vertices of a mixed graph such that, if two vertices are joined by an edge, then their colors have to be different, and if two vertices are joined by an arc, then the color of the startvertex has to be not greater than the color of the endvertex. We discuss some algorithms for coloring the vertices of a mixed graph with a small number t of colors and present computational results for calculating the chromatic number, i.e., the minimal possible value of such a t . We also study the chromatic polynomial of a mixed graph which may be used for calculating the number of feasible schedules. 相似文献
68.
Kinkar Ch. Das 《Graphs and Combinatorics》2007,23(6):625-632
Let G = (V,E) be a simple graph with n vertices, e edges and d1 be the highest degree. Further let λi, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ2 = λ3 = ... = λn-1 if and only if G is a complete graph or a star graph or a (d1,d1) complete bipartite graph.
Also we establish the following upper bound for the number of spanning trees of G on n, e and d1 only:
The equality holds if and only if G is a star graph or a complete graph. Earlier bounds by Grimmett [5], Grone and Merris [6], Nosal [11], and Kelmans [2] were
sharp for complete graphs only. Also our bound depends on n, e and d1 only.
This work was done while the author was doing postdoctoral research in LRI, Université Paris-XI, Orsay, France. 相似文献
70.
The independence polynomial, ω(G,x)=∑wkxk, of a graph, G, has coefficients, wk, that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which wk≠0. Little is known of less straightforward relationships between graph structure and the properties of ω(G,x), in part because of the difficulty of calculating values of wk for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages. 相似文献